due to the residual illumination of the spectator ions, which will be \(\chi \ll 1\)  for sufficiently-good addressing capabilities. This simple expression shows that the crosstalk contribution of each source of error, from off-resonant carrier to thermal phonon populations, has the same relative weight as discussed at length for the MS gate without crosstalk [1]}. All these contributions can then be used to extract the error rate \(p^{\rm ct}_{\rm MS}\)  that should be fed into a particular channel, such as a depolarising channel taking into account the crosstalk. For instance, if we restrict to a depolarising channel combining single-qubit and two-qubit operations [1]}, considering now both active and spectator ions, the corresponding quantum channel is
\({\varepsilon _{\rm MS}}(\rho )&=(1-p_{\rm MS}-p_{\rm MS}^{\rm ct})\rho \nonumber \\&+\frac{p_{\rm MS}}{N_{\rm MS}}\sum _{i=1}^{N}\sum _{\alpha =1,2,3}\sigma _{i}^\alpha \rho \sigma _{i}^\alpha \nonumber \\&+\frac{p_{\rm MS}}{N_{\rm MS}}\sum _{i_1, i_2}\sum _{\alpha ,\beta }\sigma _{i_1}^\alpha \sigma _{i_2}^\beta \hspace{0.84998pt}\rho \hspace{0.84998pt}\sigma _{i_1}^\alpha \sigma _{i_2}^\beta \nonumber \\&+\frac{p_{\rm MS}^{\rm ct}}{N_{\rm ct}}\sum _{j=1}^{M}\sum _{\alpha =1,2,3}\sigma _{j}^\alpha \rho \sigma _{j}^\alpha \nonumber \\&+\frac{p_{\rm MS}}{N_{\rm MS}}\sum _{j_1, j_2}\sum _{\alpha ,\beta }\sigma _{j_1}^\alpha \sigma _{j_2}^\beta \hspace{0.84998pt}\rho \hspace{0.84998pt}\sigma _{j_1}^\alpha \sigma _{j_2}^\beta ,\) 
